Print all subsequences of a string (Iterative Method)

Software Development
#1

Hello Everyone,

Given a string s, print all possible subsequences of the given string in an iterative manner.

Examples:

Input : abc Output : a, b, c, ab, ac, bc, abc Input : aab Output : a, b, aa, ab, aab

Approach 1 :

Here, we discuss much easier and simpler iterative approach which is similar to Power Set. We use bit pattern from binary representation of 1 to 2^length(s) – 1.

input = “abc”
Binary representation to consider 1 to (2^3-1), i.e 1 to 7.
Start from left (MSB) to right (LSB) of binary representation and append characters from input string which corresponds to bit value 1 in binary representation to Final subsequence string sub.

Example:
001 => abc . Only c corresponds to bit 1. So, subsequence = c.
101 => abc . a and c corresponds to bit 1. So, subsequence = ac.
binary_representation (1) = 001 => c
binary_representation (2) = 010 => b
binary_representation (3) = 011 => bc
binary_representation (4) = 100 => a
binary_representation (5) = 101 => ac
binary_representation (6) = 110 => ab
binary_representation (7) = 111 => abc

Below is the implementation of above approach:

// C++ program to print all Subsequences

// of a string in iterative manner

#include <bits/stdc++.h>

using namespace std;

// function to find subsequence

string subsequence(string s, int binary, int len)

{

string sub = "" ;

for ( int j = 0; j < len; j++)

// check if jth bit in binary is 1

if (binary & (1 << j))

// if jth bit is 1, include it

// in subsequence

sub += s[j];

return sub;

}

// function to print all subsequences

void possibleSubsequences(string s){

// map to store subsequence

// lexicographically by length

map< int , set<string> > sorted_subsequence;

int len = s.size();

// Total number of non-empty subsequence

// in string is 2^len-1

int limit = pow (2, len);

// i=0, corresponds to empty subsequence

for ( int i = 1; i <= limit - 1; i++) {

// subsequence for binary pattern i

string sub = subsequence(s, i, len);

// storing sub in map

sorted_subsequence[sub.length()].insert(sub);

}

for ( auto it : sorted_subsequence) {

// it.first is length of Subsequence

// it.second is set<string>

cout << "Subsequences of length = "

<< it.first << " are:" << endl;

for ( auto ii : it.second)

// ii is iterator of type set<string>

cout << ii << " " ;

cout << endl;

}

}

// driver function

int main()

{

string s = "aabc" ;

possibleSubsequences(s);

return 0;

}

Output:

Subsequences of length = 1 are: a b c Subsequences of length = 2 are: aa ab ac bc Subsequences of length = 3 are: aab aac abc Subsequences of length = 4 are: aabc

Time Complexity : O(2^{n} * l) , where n is length of string to find subsequences and l is length of binary string.

Approach 2 :
Approach is to get the position of rightmost set bit and and reset that bit after appending corresponding character from given string to the subsequence and will repeat the same thing till corresponding binary pattern has no set bits.

If input is s = “abc”
Binary representation to consider 1 to (2^3-1), i.e 1 to 7.
001 => abc . Only c corresponds to bit 1. So, subsequence = c
101 => abc . a and c corresponds to bit 1. So, subsequence = ac.
Let us use Binary representation of 5, i.e 101.
Rightmost bit is at position 1, append character at beginning of sub = c ,reset position 1 => 100
Rightmost bit is at position 3, append character at beginning of sub = ac ,reset position 3 => 000
As now we have no set bit left, we stop computing subsequence.

Example :
binary_representation (1) = 001 => c
binary_representation (2) = 010 => b
binary_representation (3) = 011 => bc
binary_representation (4) = 100 => a
binary_representation (5) = 101 => ac
binary_representation (6) = 110 => ab
binary_representation (7) = 111 => abc

Below is the implementation of above approach :

// C++ code all Subsequences of a

// string in iterative manner

#include <bits/stdc++.h>

using namespace std;

// function to find subsequence

string subsequence(string s, int binary)

{

string sub = "" ;

int pos;

// loop while binary is greater than 0

while (binary>0)

{

// get the position of rightmost set bit

pos=log2(binary&-binary)+1;

// append at beginning as we are

// going from LSB to MSB

sub=s[pos-1]+sub;

// resets bit at pos in binary

binary= (binary & ~(1 << (pos-1)));

}

reverse(sub.begin(),sub.end());

return sub;

}

// function to print all subsequences

void possibleSubsequences(string s){

// map to store subsequence

// lexicographically by length

map< int , set<string> > sorted_subsequence;

int len = s.size();

// Total number of non-empty subsequence

// in string is 2^len-1

int limit = pow (2, len);

// i=0, corresponds to empty subsequence

for ( int i = 1; i <= limit - 1; i++) {

// subsequence for binary pattern i

string sub = subsequence(s, i);

// storing sub in map

sorted_subsequence[sub.length()].insert(sub);

}

for ( auto it : sorted_subsequence) {

// it.first is length of Subsequence

// it.second is set<string>

cout << "Subsequences of length = "

<< it.first << " are:" << endl;

for ( auto ii : it.second)

// ii is iterator of type set<string>

cout << ii << " " ;

cout << endl;

}

}

// driver function

int main()

{

string s = "aabc" ;

possibleSubsequences(s);

return 0;

}

Output:

Subsequences of length = 1 are: a b c Subsequences of length = 2 are: aa ab ac bc Subsequences of length = 3 are: aab aac abc Subsequences of length = 4 are: aabc

Time Complexity: O(2^{n} * b) , where n is the length of string to find subsequence and b is the number of set bits in binary string.